QCD Laplace sum rules are used to calculate heavy quarkonium (charmonium and bottomonium) hybrid masses in several distinct $J^{PC}$ channels. Previous studies of heavy quarkonium hybrids did not include the effects of dimension-six condensates, lead
ing to unstable sum rules and unreliable mass predictions in some channels. We have updated these sum rules to include dimension-six condensates, providing new mass predictions for the spectra of heavy quarkonium hybrids. We confirm the finding of other approaches that the negative-parity $J^{PC}=(0,1,2)^{-+},,1^{--}$ states form the lightest hybrid supermultiplet and the positive-parity $J^{PC}=(0,1)^{+-},,(0,1,2)^{++}$ states are members of a heavier supermultiplet. Our results disfavor a pure charmonium hybrid interpretation of the $X(3872)$, in agreement with previous work.
